Chapter 3.6, Problem 43PFA

a.

To determine

To determine the function to represent the sequence.

Answer to Problem 43PFA

  $3n+9$

Explanation of Solution

Given:

Sequence:

  $12,15,18,\mathrm{21...}$

Formula used:

nth term of any arithmetic progression is given by the formula:

  $t_n=a+(n-1)d$

  $t_n=$ nth term

  $a=$ The first term of the arithmetic progression

  $d=$ The common difference of the arithmetic progression

Calculation:

The common difference of the arithmetic progression is:

  $15-12=3$

Applying above formula,

  $\Rightarrow a_n=a+(n-1)d$

  $\Rightarrow a_n=12+(n-1)(3)$  $[\because a=12\&d=3]$

  $\begin{array}{l}\Rightarrow a_n=12th+3n-3\\ \Rightarrow a_n=45\end{array}$

The required function to represent the sequence is $f(n)=3n+9$ .

Conclusion:

Hence, the missing part is $3n+9$ .

b.

To determine

To determine the number of additional sweatshirts sold at each game.

Answer to Problem 43PFA

  $3$

Explanation of Solution

Given:

Sequence:

  $12,15,18,\mathrm{21...}$

Calculation:

The common difference of the arithmetic progression is:

  $15-12=3$

Here, the common difference represents the number of additional sweatshirts sold at each game.

Conclusion:

Hence, the required number of additional sweatshirts sold at each game is $3$ .

c.

To determine

To graph the function and determine the domain and range

Answer to Problem 43PFA

  $D=\{1,2,3,\mathrm{4...}\}\&R=\{12,15,18,21,\mathrm{...}\}$

Explanation of Solution

Given:

Sequence:

  $12,15,18,\mathrm{21...}$

Calculation of Graph:

The required function to represent the sequence is $f(n)=3n+9$ .

  $(n,a_n)$ represent the points of the graph.

First four points are: $(1,12),(2,15),(3,18),(4,21)$

Graph:

  

Interpretation:

From the graph it can be observed that the horizontal extent of the graph is from $-\infty$ to $\infty$ .

And the vertical extent of the graph is also from $-\infty$ to $\infty$ .

Conclusion:

So, the domain and range of the function of the sequence are respectively:

  $D=\{1,2,3,\mathrm{4...}\}\&R=\{12,15,18,21,\mathrm{...}\}$

d.

To determine

To fund the number of game on which 45 sweat shirts are sold.

Answer to Problem 43PFA

  $12th$ game.

Explanation of Solution

Given:

Sequence:

  $12,15,18,\mathrm{21...}$

Formula used:

nth term of any arithmetic progression is given by the formula:

  $t_n=a+(n-1)d$

  $t_n=$ nth term

  $a=$ The first term of the arithmetic progression

  $d=$ The common difference of the arithmetic progression

Calculation / Explanation:

The common difference of the arithmetic progression:

  $15-12=3$

Let, the number of the game is n .

Then, $a_n=45$

Applying above formula,

  $\Rightarrow a_n=a+(n-1)d$

  $\Rightarrow a_n=12+(n-1)(3)$  $[\because a=12\&d=3]$

  $\begin{array}{l}\Rightarrow 45=12+3n-3[\because a_n^{}=45]\\ \Rightarrow 45=3n+9\\ \Rightarrow 3n=45-9\\ \Rightarrow 3n=36\\ \Rightarrow n=12\end{array}$

Conclusion:

Hence, the boosters sold $45$ shirts on $12th$ game.